He can vary x' and t' independently. If he keeps t' constant then ##\partial\tau/\partial t'## does not change - but he's still free to vary x'. That tells him that ##\partial\tau/\partial x'## is constant and independent of t' - in other words ##\tau=Ax'+f (t')##. He can make the same argument the other way around to get the dependence on t'. So the only possible solution is of the form ##\tau=Ax'+Bt'##. He then just substitutes that solution in to get the ratio of A to B, and chooses that he wants the constant a to be dimensionless.

To solve an equation of the form
$$
\frac{\partial\tau}{\partial x'}+b\frac{\partial\tau}{\partial t} = 0
$$
where ##b## is a constant, you need the change the variables to ##\xi=x'+bt## and ##\eta=x'-bt##. Then the equation becomes very simple and you can solve it. The general solutions is
$$
\tau = F(x'-bt)
$$
where ##F## is any differentialble function. If it is linear, then you get the result in the paper.